Foundations — Wave function ψ — probability density - ψ - ²

Before you can use the parent note on the wave function, you need to own every mark that appears in it. We build them one at a time. Each new symbol is defined in plain words, drawn as a picture, and justified — no symbol is used before it is earned. We will not even write the wave-function symbol until we have earned it in Section 8.

1. A function $f(x)$ — a machine that eats a number and spits out a number

The picture. Think of a graph. The horizontal axis is the input $x$ (a position on a line). The vertical axis is the output $f(x)$ (how tall the curve is above that spot). A tall curve = big output; a flat-on-the-floor curve = zero output.

Why the topic needs it. The whole subject turns out to be one special function. It takes two inputs — a position $x$ and a time $t$ (a moment on the clock) — and returns "how much wave" is present at that place and moment. We will build that function's symbol only after its ingredients are in place; for now just be comfortable that a function is a height-above-a-spot. If "function" feels shaky, look at the curve above and read off a few heights before moving on.

2. The imaginary unit $i$ — a number whose square is $-1$

Ordinary numbers, when squared, are never negative: $3^2=9$, $(-3{)}^2=9$. So the question "which number squares to $-1$?" has no answer among the numbers you know. Mathematicians invented one and named it $i$.

Why invent it? Because it unlocks a bigger number line — see the next section. Later we will find our special function is complex-valued, and complex numbers are built out of $i$. (More depth in Complex Numbers.)

3. A complex number $a+ib$ — a point on a flat plane

The picture — this is the key mental image. A single real number lives on a line. A complex number needs a whole plane: go $a$ steps right (real axis) and $b$ steps up (imaginary axis). So $z$ is an arrow from the origin to the point $(a,b)$.

Two things about that arrow matter enormously:

• its length — how far the point is from the origin,
• its direction — which way the arrow points.

Both get their own symbols below.

Why the topic needs it. The special function we are building carries, at every position, an arrow like $z=a+ib$ rather than just a plain height. The arrow's length will become probability; its direction will become phase (momentum information).

4. The complex conjugate $z^{\ast }$ — flip the arrow over the real axis

The picture. Reflecting the arrow across the horizontal (real) axis. The point $(a,b)$ becomes $(a,-b)$ — same distance from the origin, mirror-image direction.

Why we care — the magic product. Multiply a number by its own conjugate: $z^{\ast }z=(a-ib)(a+ib)=a^2-(ib{)}^2=a^2-i^2{b}^2=a^2+b^2.$ The $i$ vanishes! We are left with a plain, non-negative real number. This is the trick the whole topic hinges on: multiplying a complex value by its own conjugate throws away the imaginary weirdness and leaves something a probability can be.

5. The modulus $|z|$ — the length of the arrow

The picture. Look back at figure s02: the arrow, together with its horizontal shadow $a$ and vertical shadow $b$, makes a right triangle. The arrow is the hypotenuse, so Pythagoras gives $|z|=\sqrt{a^2+b^2}$. This is why we need a right triangle here — it's the shortest path from "coordinates $a,b$" to "length".

Connecting to the conjugate. Notice $|z{|}^2=a^2+b^2=z^{\ast }z.$ So $|z{|}^2$ and $z^{\ast }z$ are the same thing: both mean "arrow length squared".

6. Exponentiation and the number $e$ — then the phase $e^{i\theta }$

The parent topic's oscillating factors are written $e^{i\theta }$ (and $e^{ikx}$). Before we can trust that symbol we must ask two honest questions: what is $e$? and what could it possibly mean to raise it to an imaginary power?

Why this matters. The series above accepts any $u$ we plug in, including $u=i\theta$. So "$e$ to an imaginary power" is not mystical — it is just this sum with $i\theta$ in place of $u$. Let us actually do it.

The picture. $\cos \theta$ is the horizontal shadow and $\sin \theta$ the vertical shadow of a point on a circle of radius 1. So $e^{i\theta }$ is simply an arrow of length exactly 1, pointing at angle $\theta$ around the origin.

Why this shape matters for probability. Its length is $|e^{i\theta }|=\sqrt{{\cos }^2\theta +{\sin }^2\theta }=\sqrt{1}=1.$ So multiplying an arrow by $e^{i\theta }$ rotates it but never changes its length. Since probability will be length-squared, this rotation is invisible to it: $|e^{i\theta }z{|}^2=|e^{i\theta }{|}^2|z{|}^2=1\cdot |z{|}^2=|z{|}^2.$ That single fact is why "global phase doesn't change physics" in the parent note.

7. The integral ${\int }_a^{b}f(x)dx$ — the area under the curve

The parent note's core condition sums a density over all space. What is that long "S"?

The picture — why it must be a sum. Slice the region into thin vertical strips, each of width $dx$ (a tiny sliver of horizontal distance). One strip at position $x$ has height $f(x)$, so its area is $f(x)dx$. Add up every strip and you get the whole area. The $\int$ symbol is literally a stretched "S" for sum.

Why the topic needs it. The density we care about is a height $f(x)$ measuring probability per unit length. One strip $f(x)dx$ is the tiny probability of the particle landing in that sliver. Summing all strips over the whole line, from $-\infty$ (infinitely far left) to $+\infty$ (infinitely far right), must give exactly $1$: the particle is definitely somewhere.

8. Putting the marks together — earning $\psi$, and reading $|\psi {|}^2={\psi }^{\ast }\psi$

Now every ingredient is in hand, so we may finally name the special function. The parent note calls it $\psi$ (the Greek letter "psi"): a function of position $x$ and time $t$ whose output at each place is a complex number — an arrow $a+ib$, as in Section 3. Read the parent's central line left to right:

Because a global phase $e^{i\theta }$ (Section 6) has length 1, it drops out of $|\psi {|}^2$ — the physics never sees it.

Prerequisite map

The diagram below reads top-to-bottom: each arrow means "you need the upper box to understand the lower box." Everything funnels into probability density and then normalization — the two beating hearts of the parent topic.

Equipment checklist

Cover the right-hand side and test yourself. If any answer is fuzzy, reread that section before opening the parent note.